# cssExptsMult -- the exponents and multiplicities of the canonical series solutions

## Synopsis

• Usage:
cssExptsMult(I,w)
• Inputs:
• I, an ideal, (regular) holonomic ideal in the Weyl algebra
• w, a list, (generic) weights for $I$, of length half the number of variables in the Weyl algebra
• Outputs:
• a list, of exponents of the starting exponents of the canonical series solutions of I in the direction of (-w,w), as in [SST, Theorem 2.3.11], together with their multiplicities.

## Description

There are examples in the tutorial that can be moved here.

 i1 : R1 = QQ[z] o1 = R1 o1 : PolynomialRing i2 : W1 = makeWA R1 o2 = W1 o2 : PolynomialRing, 1 differential variable(s) i3 : a=1/2 1 o3 = - 2 o3 : QQ i4 : b=3 o4 = 3 i5 : c=5/3 5 o5 = - 3 o5 : QQ i6 : J = ideal(z*(1-z)*dz^2+(c-(a+b+1)*z)*dz-a*b) -- the Gauss hypergeometric equation, exponents 0, 1-c 2 2 2 9 5 3 o6 = ideal(- z dz + z*dz - -z*dz + -dz - -) 2 3 2 o6 : Ideal of W1 i7 : cssExpts(J,{1}) 2 o7 = {{0}, {- -}} 3 o7 : List i8 : c=1 -- Now we have a single exponent of multiplicity 2 o8 = 1 i9 : J = ideal(z*(1-z)*dz^2+(c-(a+b+1)*z)*dz-a*b) 2 2 2 9 3 o9 = ideal(- z dz + z*dz - -z*dz + dz - -) 2 2 o9 : Ideal of W1 i10 : cssExpts(J,{1}) o10 = {{0}} o10 : List i11 : cssExptsMult(J,{1}) o11 = {{2, {0}}} o11 : List

## Ways to use cssExptsMult :

• cssExptsMult(Ideal,List)

## For the programmer

The object cssExptsMult is .